Trigonometry Angles--Pi/7

Trigonometric functions of for an integer cannot be expressed in terms of sums, products, and finite root extractions on real rational numbers because 7 is not a Fermat prime. This also means that the heptagon is not a constructible polygon.

However, exact expressions involving roots of complex numbers can still be derived either using the trigonometric identity

(1)

with or by expressing in terms of complex exponentials and simplifying the resulting expression. Letting denote the th root of the polynomial using the ordering of the Wolfram Language's Root function gives the following algebraic root representations for trigonometric functions with argument ,

			(2)
			(3)
			(4)
			(5)
			(6)
			(7)

with argument ,

			(8)
			(9)
			(10)
			(11)
			(12)
			(13)

and with argument ,

			(14)
			(15)
			(16)
			(17)
			(18)
			(19)

Root and Galois-minimal expressions can be obtained using Wolfram Language code such as the following:

  RootReduce[TrigToRadicals[Sin[Pi/7]]]
  Developer`TrigToRadicals[Sin[Pi/7]]

Combinations of the functions satisfy

			(20)
			(21)
			(22)

(Bankoff and Garfunkel 1973). A sum identity is given by

(23)

Another interesting identity is given by

(24)

(Borwein and Bailey 2003, p. 77).